This September I was a competitor in the Mental Calculation World Cup, and much of what I learnt there about the mind and skills may be interesting for a wide audience, including readers of Sebastian's blog.
To set the scene, it’s the world’s toughest arithmetic competition, held every 2 years. Around 30-40 people from all around the world qualify to compete, some of whom dedicate much of their lives to arithmetic, while others – such as myself – have careers in other fields, such as software or education.
Usually when people hear about mental gymnastics, such as arithmetic or memory, the story is about some particular savant with a single-minded obsession for numbers. So I was pleasantly surprised to find that many of the entrants were exactly the opposite, with many interests, almost treating life like one giant party.
About a year ago, I stumbled upon this competition – and some past scoreboards – after an idle Google search, and wondered whether I could train myself up to a similar standard. At first, it seemed unlikely – I was making too many mistakes. One of the tasks is to solve multiplications of the form: 84505395 x
29817723, without writing down anything except for the answer. If you make an error in any of the one hundred and twenty-seven steps, then your answer will be wrong. Perhaps I needed to concentrate better?
Another calculation event is to determine the correct day of the week for randomly selected dates, such as 26-05-1634. The top performers in the competition solve 20-60 in a minute, while I could barely manage 10 even after practising. Could I improve if I trained a different way?
So as I began to practise, one thing that really stood out to me was how effective deliberate practice is – and this is true for very many skills such as music, language and sport – not just mental calculation. The mind learns very little from repeating an action each additional time, as it will receive very little new feedback. The neural pathways representing the actions or thoughts may become slightly stronger or slightly more available, but after they are reasonably established, there is little else to be gained by repetition.
It is more efficient instead to experiment – identifying potential weaknesses and overcoming each one. A professional musician, for example, will experiment with different ways of playing the same passages of music to discover how they can produce the best possible sound. Parts of the performance that are particularly awkward can be practised in isolation to make them feel natural.
And when I found it difficult to solve more than 10-12 calendar dates per minute, I looked for the slowest part of the process and investigated what I could do to speed it up. The result? I found a handful of values that I often had to spend considerable time calculating, and simply memorised them. Soon I was managing over 20 dates per minute.
So if there is anything that you are trying to get better at, I suggest taking the perspective of a “brain engineer” and routinely look for small ways that you could program your mind to do the task better. This could be memorising some useful information, reducing how often you have to make decisions by choosing strategies in advance, or improving how easily you notice certain specific details that you otherwise often miss.
Plus for most activities, experimenting is more interesting than mindless repetition!
When engaging in an intense mental task such as arithmetic, you can get a clear sense of what is possible – and not possible – with the mind. Whilst taking the cube root of a number, it is necessary to store rather a lot of numbers in your short-term memory all at the same time. Some of them get stored as spoken words, in a kind of loop of a few seconds of sound. However, this loop is quite short – barely enough for a telephone number.
The other place where the short-term memory can store information is as a vague image, although this fades fast unless it is used immediately. I found that this vague image is where each small step of calculation actually happens, such as 4+9, whilst the loop of sounds would store other numbers for later use. Some of the Asian mental calculators train to be able to visualise an abacus very accurately, and use this to speed up their additions.
When you need to use any of the data in the loop, you have to wait a second or so until you hear it again. In the competition I would be racing against the clock, so to avoid wasting all these precious fractions of seconds, I would store numbers in a more concise format where possible. For example “two hundred and thirty-four and a half” would be stored as “two-three-four-chh” where the “chh” represents the “and a half”.
Training for a specific skill such as a mental calculation task can bring many other benefits. A major one is improved concentration, and improved control over your mind – similar to meditation. Another is the opportunity to see first-hand how a skill develops through various degrees of proficiency – especially if there is an easy way to measure your ability at any time, as there is with arithmetic tasks.
If you want to learn more about the learning process, pick a skill – especially one that’s easy to measure – and try getting as good at is as you can as fast as you can, noting what works and what does not work. Maybe you could try learning to write with your opposite hand, or typing faster, or sight-reading music, or even try out arithmetic like I did.
You might also be wondering about the results of the competition: how good are the very best mental calculators, and how was my performance in comparison? I came 8th out of 28 in cube roots, 13th in square roots, 13th in calendar dates (18 dates) and placed in the bottom half in addition and multiplication.
The overall winner was an abacus teacher from Japan called Naofumi Ogasawara, who came first in addition and square roots, and won all 5 “surprise” rounds. During the event he also set 4 world records, including for adding together 5-digit numbers flashed on a screen: https://www.youtube.com/watch?v=cC3PhLWwiCM
Freddis Hernández is the Cuban champion, world champion and record-holder in multiplication, and won the Memoriad Trophy in the 2012 competition. To give an idea of his speed, he can solve the problems such as “84505395 x 29817723” in under 24 seconds without writing down any working and usually without any mistake!
Myagmasuran Tuuruul, from Mongolia, won the calendar dates event, with 57 correct in one minute. Yusnier Viera, currently holds the world record, with 93! Here is a video of him achieving 90 in front of a live audience: http://youtu.be/HG9M_hRAOwQ
Full results are available here: http://www.recordholders.org/en/events/worldcup/2012/results.html
Memorization techniques? I used spaced repetition to learn the squares from 1x1=1 to 99x99=9801, and then wrote a simple program in Matlab which I could use to practise recalling those answers from memory very quickly. In the algorithm I use for calendar dates, one stage involves getting the year number (for example in 2013 it is "13") and doing a few computations on that, so I memorized the answers (probably not as well as I could have, to be honest) and used a similar program to practise rapid recall.
Visualisation techniques? Here I didn't try anything specific as it was only after the competition and meeting some of the other competitors that I realised that improved visualisation skills could help a lot for some events - especially the events where you need to hold in your mind a lot of information at once. Sometimes I found I would need to wait a moment until both the information held as sound and the image being imagined were clear enough to continue, which of course would be a disadvantage in a competition.
I have been wondering about this though, so I'll pass this particular question back over to you :p have you found any particular visualization techniques to be useful?
Great post, Daniel. Do you have any advice or recommended resources for people interesting in learning fast mental calculations? I've spent a lot of time becoming proficient in advanced memory techniques and it has been tremendously useful for me. Adding fast calculations to my mental toolbox would provide a lot of utility to daily life as well. I'm particularly interested in dates, and fast multiplication/division (with a max of 4-digit x 4-digit). I've browsed the web looking at some of the various algorithms used to calculate dates, for example, and if I recall correctly there are many different options - depending on big the range of years is.
Also, I'm curious, do mental arithmetic techniques require you to use memory techniques (ex. method of Loci) as well? You mentioned the 8-digit x 8-digit requires 127 steps to complete the calculation.
Hey Jeff - I've read before about Method of Loci & pegging systems for memorization, although never tried them myself. The information that I wanted to memorize was small enough that I could simply learn it using spaced repetition without a more advanced system in place. Also, the key thing in calculation competitions is to be able to recall the information immediately, with as few intermediate steps as possible - so while a memory palace may help learn the information faster, ultimately you'd just want to be able to respond to the data "67" with either "4489" or "6"/"-1" depending whether you're doing square roots or calendar dates.
I do find however that it can speed up learning to have some understanding of why the answer arises (and this is true for learning vocab, or anything else), and this also helps if you forget an answer, because you can quickly find other information which will remind you of the answer. For example: if I want to square "67" and have forgotten the answer, I know that 67 is about 2/3 of 100, so 67 squared is about 4/9 of 10,000, so I'm looking for an answer that is 4444ish. From there it's easy to remember the last two digits to get 4489.
The 127-step method for multiplication is actually completely straightforward (it's called the cross-method) and won't take you any time to learn :) For 4-digit x 4-digit you only need 33 calculations.
Did you know that there are competitions in memory sports too? Check out Memoriad (which also does mental calculation) and the World Memory Championship.
Finally, the algorithm for calendar dates? The best methods are variations on the Doomsday algorithm, where essentially you get a contribution from the year, century, month and day and add them together, taking the remainder when you divide by 7. The contribution from the year is complicated to calculate, but can be memorised.
Yusnier Viera has a short video explaining it here: https://www.youtube.com/watch?v=izxcziWnArg
In my method, the contributions for the months are 1 less, the contributions for the centuries are 2 more, and I don't add the 1 at the end so it amounts to the same thing.
I travel a lot.
Not all countries have the same standardization and cash controls as the United States or Western Europe. In most of the world, actually, everything is pretty ad hoc.
Part of this means, at least 5-10 times per year I'm having someone hand me the wrong amount of change or otherwise screwing up billing pricing.
Funny enough, no one ever hands me too much change. No one ever accidentally marks a bill as paid that isn't paid.
But the reverse happens. Often.
Chinese are considered to be the inventors of Abacus during 500 BC and the most primitive written document on Abacus is in Chinese that dates back in the 2nd century BC